We introduce and study some families of groups whose irreducible characters take values on quadratic extensions of the rationals. We focus mostly on a generalization of inverse semi-rational groups, which …
We introduce and study some families of groups whose irreducible characters take values on quadratic extensions of the rationals. We focus mostly on a generalization of inverse semi-rational groups, which we call uniformly semi-rational groups. Moreover, we associate to every finite group two invariants, called rationality and semi-rationality of the group. They measure respectively how far a group is from being rational and how much uniformly rational it is. We determine the possible values that these invariants may take for finite nilpotent groups. We also classify the fields that can occur as the field generated by the character values of a finite nilpotent group.
A finite group $G$ is called uniformly semi-rational if there exists an integer $r$ such that the generators of every cyclic sugroup $\langle x \rangle$ of $G$ lie in at …
A finite group $G$ is called uniformly semi-rational if there exists an integer $r$ such that the generators of every cyclic sugroup $\langle x \rangle$ of $G$ lie in at most two conjugacy classes, namely $x^G$ or $(x^r)^G$. In this paper, we provide a classification of uniformly semi-rational non-abelian simple groups with particular focus on alternating groups.
Let $G$ be a finite group and, for a given complex character $\chi$ of $G$, let ${\mathbb{Q}}(\chi)$ denote the field extension of ${\mathbb{Q}}$ obtained by adjoining all the values $\chi(g)$, …
Let $G$ be a finite group and, for a given complex character $\chi$ of $G$, let ${\mathbb{Q}}(\chi)$ denote the field extension of ${\mathbb{Q}}$ obtained by adjoining all the values $\chi(g)$, for $g\in G$. The group $G$ is called quadratic rational if, for every irreducible complex character $\chi\in{\rm{Irr}}(G)$, the field ${\mathbb{Q}}(\chi)$ is an extension of ${\mathbb{Q}}$ of degree at most $2$. Quadratic rational groups have a nice characterization in terms of the structure of the group of central units in their integral group ring, and in fact they generalize the well-known concept of a cut group (i.e., a finite group whose integral group ring has a finite group of central units). In this paper we classify the Frobenius groups that are quadratic rational, a crucial step in the project of describing the Gruenberg-Kegel graphs associated to quadratic rational groups. It turns out that every Frobenius quadratic rational group is uniformly semi-rational, i.e., it satisfies the following property: all the generators of any cyclic subgroup of $G$ lie in at most two conjugacy classes of $G$, and these classes are permuted by the same element of the Galois group ${\rm{Gal}}({\mathbb{Q}}_{|G|}/{\mathbb{Q}})$ (in general, every cut group is uniformly semi-rational, and every uniformly semi-rational group is quadratic rational). We will also see that the class of groups here considered coincides with the one studied in [4], thus the main result of this paper also completes the analysis carried out in [4].
An element x of a finite group G is called rational if all generators of the group 〈x〉 are contained in a single conjugacy class. If all elements of G …
An element x of a finite group G is called rational if all generators of the group 〈x〉 are contained in a single conjugacy class. If all elements of G are rational, then G itself is called rational. It was proved by Gow that if G is a rational solvable group then π(G) ⊂ {2, 3, 5}. We call x ∈ G semi-rational if all generators of 〈x〉 are contained in a union of two conjugacy classes. Furthermore, we call x ∈ G inverse semi-rational if every generator of 〈x〉 is conjugate to either x or x–1. Then G is called semi-rational (resp. inverse semi-rational) if all elements of G are semi-rational (resp. inverse semi-rational). We show that if G is semi-rational and solvable then π(G) ⊂ {2, 3, 5, 7, 13, 17}, and if G is inverse semi-rational and solvable then 17 ∉ π(G). If G has odd order, then it is semi-rational if and only if it is inverse semi-rational. In this case we describe the structure of G.
The aim of this article is to explore global and local properties of finite groups whose integral group rings have only trivial central units, so-called cut groups. For such a …
The aim of this article is to explore global and local properties of finite groups whose integral group rings have only trivial central units, so-called cut groups. For such a group we study actions of Galois groups on its character table and show that the natural actions on the rows and columns are essentially the same, in particular the number of rational-valued irreducible characters coincides with the number of rational-valued conjugacy classes. Further, we prove a natural criterion for nilpotent groups of class 2 to be cut and give a complete list of simple cut groups. Also, the impact of the cut property on Sylow 3-subgroups is discussed. We also collect substantial data on groups which indicates that the class of cut groups is surprisingly large. Several open problems are included.